Studying basin bifurcations in nonlinear triopoly games by using 3D visualization

We consider three-dimensional discrete dynamical system, obtained by the it eration of a noninvertible map of R-3 which simulates the time evolution of an oligopoly game with three competing firms. The model is characterized b y the presence of several coexisting stable equilibria, each with its own b asin of attraction. In this paper we face the question of the delimitation of the basins and the detection of the global bifurcations that cause the c reation of non-connected basins. This requires a study of the global proper ties of the 3-dimensional noninvertible map by the method of critical sets, based on the determination of the contact bifurcations through a systemati c computer-assisted study. This requires the visualization of Surfaces (the critical surfaces and the basins' boundaries) which sometimes are nested o ne inside the other. Enhanced graphical methods, based on two-level volume rendering, are employed in order to modulate the opacity of outer objects s o that the contacts between the basins' boundaries and critical surfaces ca n be visualized. This is obtained through the realization of ad-hoc routine s, which allow interactive 3D visualization.